Ideally, the sparsity of interactions, commutativity of the Hamiltonian, and prior knowledge about the initial state can all be combined to yield an even faster digital quantum simulation. This combination, however, appears to be technically challenging to achieve. Indeed, the state-of-the-art analysis of Trotterization represents the simulation error in terms of nested commutators of Hamiltonian terms with exponential conjugations [1]}. This representation is versatile for computing the commutator scaling of Trotter error, but it yields little information about the energy of the initial state. To the best of our knowledge, the only previous attempt to address this problem was made by Somma for simulating bosonic Hamiltonians [2]}, whose solution seems to have a divergence issue.Recent work [3]} developed a tighter analysis of Trotter error for time-dependent Hamiltonian simulation that exploits both commutativity of the target Hamiltonian and knowledge about the initial state. Instead, we combine the sparsity, the commutativity and the initial-state information to give an improved simulation of a class of interacting electrons.
